Continuous Time Galerkin Method Research Articles

An Overview of High-Order Implicit Algorithms for First-/Second-Order Systems and Novel Explicit Algorithm Designs for First-Order System Representations

In this paper, we are interested in high-order algorithms for time discretization and focus upon the high-order implicit/explicit algorithm designs. Five high-order unconditionally stable implicit algorithms, derived by the time continuous Galerkin method, weighting parameter method, collocation method, differential quadrature method, and the modified time-weighted residual method, in first-/second-order transient systems are taken into consideration. The present overview and contributions encompass: (1) The pros and cons of various methodologies for the design of high-order algorithms are first demonstrated. Generally, p unknown variables leads to the optimized $$(2p-1)$$ th-order accurate algorithms with controllable numerical dissipation, and/or 2pth-order accurate algorithms without controllable numerical dissipation. (2) Although it is claimed that the TCG method can achieve 2pth-order accuracy with controllable numerical dissipation, it will be shown in this paper that the conclusion was arrived via an inconsistent analysis for the accuracy and the controllable numerical dissipation. (3) Given the rapid increase on the computational cost for high-order algorithms, the iterative predictor/multi-corrector technique is applied to show a novel design for the high-order explicit algorithms derived from the high-order implicit algorithms for the first-order transient systems. Coupled with the high-order Legendre SEM (likewise isogeometric analysis, DG methods, p-version FEM, etc., can be employed) for the spatial discretization, this newly proposed explicit numerical framework can achieve and preserve high-order accuracy in both space and time. In comparison to the famous explicit Runge-Kutta method, these newly designed explicit algorithms have better solution accuracy with comparable stability region.

Numerical analysis of a continuous Galerkin method for damped sine‐Gordon equation

AbstractIn this article, we discuss the numerical solution for the two‐dimensional (2‐D) damped sine‐Gordon equation by using a space–time continuous Galerkin method. This method allows variable time steps and space mesh structures and its discrete scheme has good stability which are necessary for adaptive computations on unstructured grids. Meanwhile, it can easily get the higher‐order accuracy in both space and time directions. The existence and uniqueness to the numerical solution are strictly proved and a priori error estimate in maximum‐norm is given without any space–time grid conditions attached. Also, we prove that if the mesh in each time level is generated in a reasonable way, we can get the optimal order of convergence in both temporal and spatial variables. Finally, the convergence rates are presented and analyzed by some numerical experiments to illustrate the validity of the scheme.

A priori $$\varvec{L^{\infty }(L^2)}$$ error estimates for finite element approximations to parabolic integro-differential equations with discontinuous coefficients

Finite element treatment for parabolic integro-differential equations with discontinuous coefficients are presented in this work. Due to low global regularity of the solutions, the error analysis technique of the standard finite element method is difficult to adopt for interface problems. In this paper, convergence of continuous time Galerkin method for the spatially discrete scheme and backward difference scheme in time direction are discussed in $$L^{\infty }(L^2)$$ norm for fitted finite element method with straight interface triangles. More precisely, optimal error estimates are derived in $$L^{\infty }(L^2)$$ norm when initial data $$u_0\in H^3\cap H_0^1(\Omega )$$ . The interface is assumed to be smooth for our purpose.

Analysis of a space–time continuous Galerkin method for convection-dominated Sobolev equations

The convergence of space–time continuous Galerkin (STCG) method for the Sobolev equations with convection-dominated terms is studied in this article. It allows variable time steps and the change of the spatial mesh from one time interval to the next, which can make this method suitable for numerical simulations on unstructured grids. We prove the existence and uniqueness of the approximate solution and get the optimal convergence rates in L∞(H1) norm which do not require any restriction assumptions on the space and time mesh size. Finally, some numerical examples are designed to validate the high efficiency of the method showed herein and to confirm the correctness of the theoretical analysis.

A new space–time continuous Galerkin method with mesh modification for Sobolev equations

In this study, we first propose the continuous Galerkin (CG) method for the Sobolev equations, which allows different temporal step-sizes and spatial grids in each time step. And then, we demonstrate the existence and uniqueness of the approximate solutions and derive the optimal rates of convergence of the approximate solutions under the restrictive assumptions that the space–time finite element subspaces between two successive time steps are conforming elements. Finally, we provide some numerical examples on unstructured meshes to demonstrate the efficiency and flexibility of this method.

Accurate 3-D finite element simulation of elastic wave propagation with the combination of explicit and implicit time-integration methods

One of the big issues in finite element solutions of wave propagation problems is the presence of spurious high-frequency oscillations that may lead to divergent results at mesh refinement. The paper deals with the extension of the new two-stage time-integration technique developed in our previous papers to the solution of wave propagation problems with explicit time-integration methods.The explicit central difference method is used for accurate time-integration of the semi-discrete system of elastodynamics at the stage of basic computations and allows spurious high-frequency oscillations. To filter these oscillations, pre- or/and post-processing (the filtering stage) is applied using a few time increments of the implicit time-continuous Galerkin method with large numerical dissipation.A special calibration procedure is used for the selection of the minimum necessary amount of numerical dissipation (in terms of a time increment) at the filtering stage. In contrast to existing approaches that use a time-integration method with the same dissipation (or artificial viscosity) for all time increments, the new technique yields accurate and non-oscillatory results for wave propagation problems without interaction between user and computer code. The solutions of 3-D wave propagation and impact problems show the effectiveness of the new approach.

A new explicit predictor–multicorrector high-order accurate method for linear elastodynamics

A new explicit predictor–multicorrector high-order accurate method is suggested for linear elastodynamics. The method is derived from the implicit high-order accurate method based on the time-continuous Galerkin method proposed earlier in our papers. The basic unknowns for the method are displacements and velocities; accelerations are not calculated. The explicit method uses a predictor–multicorrector technique with one or two passes in order to reach the fourth order of accuracy and has controllable numerical dissipation for the suppression of spurious high-frequency oscillations. In contrast to recently suggested explicit high-order accurate methods based on the time-discontinuous Galerkin method, the new method is more accurate (has a higher order of accuracy) and has better algorithmic properties (e.g., a higher-stability limit) at the same computational efforts. Presented numerical examples show the performance of the new method. The method appears to be competitive for medium- and long-term analysis when accuracy of numerical solutions arises an issue due to error accumulation.

A numerical approach to space-time finite elements for the wave equation

We study a space-time finite element approach for the nonhomogeneous wave equation using a continuous time Galerkin method. We present fully implicit examples in 1+1, 2+1, and 3+1 dimensions using linear quadrilateral, hexahedral, and tesseractic elements. Krylov solvers with additive Schwarz preconditioning are used for solving the linear system. We introduce a time decomposition strategy in preconditioning which significantly improves performance when compared with unpreconditioned cases.

A priori error estimates in the finite element method for nonself-adjoint elliptic and parabolic interface problems

We derive a priori error estimates in the finite element method for nonselfadjoint elliptic and parabolic interface problems in a two-dimensional convex polygonal domain. Optimal H 1-norm and sub-optimal L 2-norm error estimates are obtained for elliptic interface problems. For parabolic interface problems, the continuous-time Galerkin method is analyzed and an optimal order error estimate in the L 2(0,T;H 1)-norm is established. Further, a discrete-in-time discontinuous Galerkin method is discussed and a related optimal error estimate is obtained. Keywords: Elliptic and parabolic interface problems, finite element method, spatially discrete scheme, discontinuous Galerkin method, error estimates Mathematics Subject Classification (1991): 65N15, 65N20

A new high-order accurate continuous Galerkin method for linear elastodynamics problems

A new high-order accurate time-continuous Galerkin (TCG) method for elastodynamics is suggested. The accuracy of the new implicit TCG method is increased by a factor of two in comparison to that of the standard TCG method and is one order higher than the accuracy of the standard time-discontinuous Galerkin (TDG) method at the same number of degrees of freedom. The new method is unconditionally stable and has controllable numerical dissipation at high frequencies. An iterative predictor/multi-corrector solver that includes the factorization of the effective mass matrix of the same dimension as that of the mass matrix for the second-order methods is developed for the new TCG method. A new strategy combining numerical methods with small and large numerical dissipation is developed for elastodynamics. Simple numerical tests show a significant reduction in the computation time (by 5–25 times) for the new TCG method in comparison to that for second-order methods, and the suppression of spurious high-frequency oscillations.

Optimal Error Estimates for Linear Parabolic Problems with Discontinuous Coefficients

A finite element discretization is proposed and analyzed for a linear parabolic problems with discontinuous coefficients. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with classical finite element methods [Numer. Math., 79 (1998), pp. 175--202]. In this paper, we have used a finite element discretization, where interface triangles are assumed to be curved triangles instead of straight triangles as in classical finite element methods. Optimal order error estimates in L2 and H1 norms are shown to hold even if the regularity of the solution is low on the whole domain. While the continuous time Galerkin method is discussed for the spatially discrete scheme, the discontinuous Galerkin method is analyzed for the fully discrete scheme. The interfaces and boundaries of the domains are assumed to be smooth for our purpose.

Error estimates for the numerical approximation of time-dependent flow of Bingham fluid in cylindrical pipes by the regularization method

The flow of a Bingham fluid in a cylindrical pipe can give rise to free boundary problems. The fluid behaves like a viscous fluid if the shear stress, expressed as a linear function of the shear rate, exceeds a yield value, and like a rigid body otherwise. The surfaces dividing fluid and rigid zones are the free boundaries. Therefore the solution for such highly nonlinear problems can in general only be obtained by numerical methods. Considerable progress has been made in the development of numerical algorithms for Bingham fluids [2,20,27,29,30,32]. However, very little research can be found in the literature regarding the rate of convergence of the numerical solution to the true continuous solution, that is the error estimate of these numerical methods. Error estimates are a critically important issue because they tells us how to control the error by appropriately choosing the grid sizes and other related parameters. This paper concerns the error estimates of a unsteady Bingham fluid modeled as a variational inequality due to Duvaut-Lions [16] and Glowinski [30]. The difficulty both in the analytical and numerical treatment of the mathematical model is due to the fact that it contains a nondifferentiable term. A common technique, called the regularization method, is to replace the non-differentiable term by a perturbed differentiable term which depends on a small regularization parameter ?. The regularization method effectively reduces the variational inequality to an equation (a regularized problem) which is much easier to cope with. This paper has achieved the following. (1) Error estimates are derived for a continuous time Galerkin method in suitable norms. (2) We give an estimate of the difference between the true solution and the regularized solution in terms of ?. (3) Some regularity properties for both regularized solution and the true solution are proved. (4) The error estimates for full discretization of the regularized problem using piecewise linear finite elements in space, and backward differencing in time are established for the first time by coupling the regularization parameter ? and the discretization parameters h and Δt. (5) We are able to improve our estimates in the one-dimensional case or under stronger regularity assumptions on the true solution. The estimates for the one-dimensional case are optimal and confirmed by numerical experiment. The estimates from (4) and (5) provide very important information on the measure of the error and give us a powerful mechanism to properly choose the parameters h, Δt and ? in order to control the error (see Corollary 4.4). The above estimates extend the error bounds derived in Glowinski, Lions and Tremolieres [32] (chapter 5, pp. 348---404) for the stationary Bingham fluid to the time-dependent one, which is the main contribution of this paper.

On direct methods for the discretization of a heat-conduction equation using spline functions

The finite element collocation method is sometimes equivalent to the Galerkin method. The two methods are therefore considered together here. The nonhomogeneous heat-conduction equation with nonhomogeneous boundary condition is chosen for consideration. The equation is transformed into a variational form by continuous time Galerkin method and expressed in terms of linear spline basis function. On the other hand, the collocation method is applied on the heat equation with cubic splines as the basis functions. The two methods of discretization transform the heat equation to systems of ordinary differential equations in time, respectively. The corresponding stiffness matrices for the two systems of equations turn out to be the same. These matrices are sparse, banded and symmetric, and are therefore convenient for computer application.

Existence of Global Weak Solutions for Coupled Thermoelasticity under Non-Linear Boundary Conditions

The existence of global weak solutions for coupled thermoelasticity with non-linear contact boundary conditions corresponding to the friction problem is considered. The time-continuous Galerkin method and a priori estimates obtained with Gronwall's inequality in connection with embedding theorems are applied to accomplish a straightforward generalization of one of the results proved by Martins and Oden 9.

Behaviour in the large of numerical solutions to one-dimensional nonlinear viscoelasticity by continuous time Galerkin methods

We analyze the long time behaviour of fully discrete solutions to a one-dimensional nonlinear viscoelastic problem. It is shown that these approximations which are found by a continuous time Galerkin method converge to a steady state. The possible numerical steady states are characterized and in particular their high degree of dependence on initial data and mesh design is explained. Computational results are included which show the above dependence and indicate that the numerical solutions will typically not converge to unstable states.

On the Numerical Solution of Parabolic Equations in a Single Space Variable by the Continuous Time Galerkin Method

We consider the Galerkin method to solve a parabolic initial boundary value problem in one space variable, using piecewise polynomial functions and give an alternative proof of superconvergence. Then by means of Lobatto quadrature, we obtain purely explicit vector initial value problems without loss in the order of accuracy, global or pointwise.

Galerkin Methods for First Order Hyperbolics: An Example

The continuous-time Galerkin method is studied for the equation $u_t + u_x = 0$ with periodic solution. If the space of possible approximate solutions is taken to be $C^1 $ piecewise cubic polynomials on mesh of size h, then the $L^2 $-norm of the error is in general no better than $ch^3 $; if the class of possible approximate solutions is taken to be $C^2 $ piecewise cubic polynomials on this mesh, the error is bounded by $ch^4 $ for sufficiently smooth solutions.